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Bit-Interleaved Coded Modulation: Fundamentals, Analysis and Design by Alex Alvarado, Leszek Szczecinski

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4.3.2 BICM Pseudo-Capacity

The results from the previous section point out the fact that binary labeling strongly affects the BICM-GMI. Moreover, we have seen that the BICM-GMI curves for a given binary labeling intersect each other for different constellation sizes, which naturally leads to a quite general question: What is the best constellation, binary labeling, and input distribution for the BICM at a given SNR? Once this question is answered, approaching the fundamental limit will depend only on a good design of the binary encoder/decoder.

To formalize the previous question, and in analogy with (4.25), we define the BICM pseudo-capacity for a given constellation size c04-math-0341as

where the optimization is over the constellation and its binary labeling, as well as over the vector of bitwise probabilities c04-math-0343 that induce a symbol-wise PMF (via (2.72)) satisfying the energy constraint c04-math-0344. We use the name BICM pseudo-capacity because, in theory, the quantity in (4.64) might not be the largest achievable rate, which simply comes from the fact that the BICM-GMI is not the largest achievable rate ...

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